Using the master theorem read off the order of the following recurrences: (a) T(n) = 2T(n/2) + n² (c) T(n) = 4T(n/2) + n (b) T(n) = 2T(n/2) +n (d) T(n) = T(n/4) +1
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- Given the following code snippet, find the following:1) Recurrence relation of func (x,y) when x>0 and length of y is n.T(x,n) = T(_) + O(_) (Just fill in the blanks)For 2-3, consider worst-case scenario and initial value of y as [1]2) Time complexity of func (x,y)3) Auxiliary Space Complexity of func (x,y)Use the Master Theorem to find the complexity class ( Θ notation ) of the following recurrences a) T(n) = 4T(n/2) + n3 b) T(n) = 4T(n/3) + n c) T(n) = 4T(n/3) + n2Find a bound for each recurrence below using Master Theorem:(a) T(n) = 3T(n=2) + n2(b) T(n) = 2T(n=4) + n0:49(c) T(n) = 3T(n=3) + n=2
- I am not an engineering student. Grateful for your detailed explanation.Give tight asymptotic bounds for the following recurrences. • T(n) = 4T(n/2) + n 3 − 1. • T(n) = 8T(n/2) + n 2 . • T(n) = 6T(n/3) + n. • T(n) = T( √ n) + 11. The Fibonacci numbers F(0), F(1), F(2), are defined by the recurrence relation:F(0) = 0,F(1) = 1,F(n) = F(n-1) + F(n-2) for n >= (greater than or equal to) 2.The first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, and 21. Prove by induction that for all n >= (greater than or equal to) 1.[F(n-1)][F(n+1)] - F(n)2 = (-1)n.Hint: Express F(n-1) in terms of F(n)and F(n-2) and express F(n+1) in terms of F(n) and F(n-1). Expand and simplify, then apply the inductive hypothesis. Handwritten onlyData Structures and Algorithms Solve the following recurrence relations. x(n) = x(n − 1) + 5 for n > 1, x(1) = 0 x(n) = x(n − 1) + n for n > 0, x(0) = 0 x(n) = x(n/2) + n for n > 1, x(1) = 1
- Given the following code snippet, find the following:1) Recurrence relation of func (x,y) when x>0 and length of y is n. Express in the form below:T(x,n) = T(?) + O(?)For 2-3, consider worst-case scenario and initial value of y as [1]2) Time complexity of func (x,y)3) Space Complexity and Auxiliary Space Complexity of func (x,y)Determine Big O for the following recurrence relations using the Master Theorem if applicable, otherwise use Repeated Substitution. Show your work. Simplify the answer. a) T(n) = 1T(n) = 9T(n/3) + n b) T(n) = 1T(n) = 2T(n - 1) + 1 Master Theorem when n=1 when n>1 when n=1 when n>1 For T(n) = A T(n/B) + O(nk) where A 3 1, B > 1It was claimed that:(a, b) ≤ (c, d) ⇔ (a < c) ∨ (a = c ∧ b ≤ d) defines a well-ordering on N x N. Show that this is actually the case.
- Let T(n) be the maximum number of guesses required to correctly identify a secret word that is randomly chosen from a dictionary with exactly n words. Determine a recurrence relation for T(n), explain why the recurrence relation is true, and then apply the Master Theorem to show that T(n) = Θ(log n).Suppose that f(n) satisfies the divide-and-conquer recurrence relation f(n) = 3f(n/4)+n2/8 with f(1) = 2. What is f(64)?Consider the following recurrence relation: T(n) = 5*(T(n/5)) + 2 with n = 5^k, and T(1) = 1 We have: T(5^0) = T(1) = 1 T(5^1) = T(5) = 5*(T(5/5)) + 2 = 5 * T(1) +2 = 5 *1 +2 = 7 T(5^2) = T(25) = 5*(T(25/5)) + 2 = 5 * T(5) +2 = 5 * 7 + 2 = 37 T(5^3) = T(125) = 5*(T(125/5)) + 2 = 5 * T(25) +2 = 5 * 37 + 2 = 187 etc....... Find T(5^k) = ?